Align object with group of vertices
I wasn't able to find a way to perfectly align the peices with a snap of sort like I had hoped. That said, I was able to find a method that allowed me to align them to a degree of precision that makes it so that even the part that should be the furthest away is aligned to match exactly to within a millionth of a unit.   


Snap one vertex


Enter Edit mode on the object to be moved (Select it and press Tab)  
Select one vertex  
Move the 3D cursor to that spot (Shift + S, Cursor to Active)
Exit Edit mode (Press Tab)
Set the origin to that point (Shift + Control +
Alt + S, Origin to 3D Cursor)   
Turn on Snap During Transform (Shift + Tab)
Set its Type Of Element To Snap To to Vertex (In the header, next to the magnet icon)
Set its Part To Snap Onto (just to the right of the Type Of Element To Snap To) to Active
Move it until it snaps (G key to move, move your mouse to the vertex that should align)
Rotate it most of the way
In this case, I knew that it was already knew it was aligned on one axis, so I rotated it along that axis (in this case the Z axis). Because the origin is now at the perfectly aligned vertex, it won't move if you set the Pivot Center to work by its origin (On the header, to the right of the mode option and the display method, set to Active Element)
I rotated along the Z axis (R to start rotating,Z` to then constrain to the Z axis). I got it decently close. 
Fine tune
I then zoomed in as close as I could get to the point that would be the furthest away (opposite side of the circle of vertexes) and started editing the object rotation while in Object mode and wireframe view mode (Z while in Object mode). You can find this in the Object section of the Properties window. I just edited the Z rotation little by little until it matched as closely as I could get. After checking, there is zero difference in the position of the vertexes now.